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Quick Study: Edward Frenkel on math

It's a lot like borscht

EDWARD FRENKEL is a Russian mathematician working in representation theory, algebraic geometry and mathematical physics. He is professor of mathematics at the University of California, Berkeley, and the author of “Love and Math”, recently published by Basic Books.

You describe math as "beautiful". What do you mean?

Imagine you had an art class in which they taught you how to paint a fence, but never showed you the great masters. Of course, you would say; ‘I hate art.' You were bad at painting the fence but you wouldn’t know what else there is to art. Unfortunately, that is exactly what happens with mathematics. What we study at school is a tiny little part of mathematics. I want people to discover the magic world of mathematics, almost like a parallel universe, that most of us aren’t aware even exists.

How did you discover it?

When I was growing up near Moscow I thought mathematics was the most boring and irrelevant subject, but I was fascinated with quantum physics and elementary particles. Luckily for me a professional mathematician was a friend of my family and when I was about 15 years old he said to me; ‘Do you know that this theory of elementary particles is based on mathematics?’ He showed me a book full of formulas and equations I could not understand, but I realised that these were glimpses of this magic world that was hidden from me and this was portal into that world. It was love at first sight. What professional mathematicians do goes to the heart of reality, to the heart of the universe. It’s what enables us to learn how the world works.

Can complex mathematical ideas be explained to someone who is not a mathematician?

Yes. Take the idea of symmetry. In what sense is a round table more symmetrical than a square table? In a round table any rotation around the centre point preserves the table’s shape and location. But if I turn a square table at a random angle you can see the difference—only four rotations, by multiples of 90 degrees, will preserve it. Transformation of an object which preserves its shape and location is what we call a symmetry. Anyone can understand this. The way physicists learnt about elementary particles and were able to theorise their existence was precisely through a theory of symmetries. The Higgs Boson is this elusive particle recently discovered by the Large Hadron Collider in Geneva, and for this prediction two physicists were awarded the Nobel prize. That prize was really awarded for mathematical prediction because the origins of that theory are in symmetry.

Symmetry exists without human beings to observe it. Does maths exist without human beings to observe it, like gravity? Or have we made it up in order to understand the physical world?

I argue, as others have done before me, that mathematical concepts and ideas exist objectively, outside of the physical world and outside of the world of consciousness. We mathematicians discover them and are able to connect to this hidden reality through our consciousness. If Leo Tolstoy had not lived we would never have known Anna Karenina. There is no reason to believe that another author would have written that same novel. However, if Pythagoras had not lived, someone else would have discovered exactly the same Pythagoras theorem. Moreover, that theorem means the same to us today as it meant to Pythagoras 2,500 years ago.

So it’s not subject to culture?

This is the special quality of mathematics. It means the same today as it will a thousand years from now. Our perception of the physical world can be distorted. We can disagree on many different things, but mathematics is something we all agree on.

The only reason the theory means the same is that it describes the reality of the physical world, so mathematics must need the physical world.

Not always. Euclidian geometry deals with flat spaces, such as the three-dimensional flat space. For millennia people thought we inhabited a flat, three-dimensional world. It was only after Einstein that we realised we lived in a curved space and that light doesn’t travel in a straight line but bends around a star. Pythagoras theorem is about geometric shapes in an idealised space, a flat Euclidian plane which, in fact, is not found in the real world. The real world is curved. When Pythagoras discovered his theorem there were, of course, inferences from physical reality, and a lot of mathematics is drawn from our experience in the physical world, but our imagination is limited and a lot of mathematics is actually discovered within the narrative of a hidden mathematical world. If you look at recent discoveries, they have no a priori bearing in physical reality at all.

The naive interpretation that mathematics comes from physical reality just doesn’t work. The other interpretation that mathematics is a product of the human mind also has serious issues, because it seems clear that some of these concepts transcend any specific individual.

Take Evariste Galois, who was killed in a duel at the age of 20. He came up with a beautiful theory on symmetry called Galois theory. His contemporaries didn’t get it but this theory now forms the core of modern mathematics. But what if the work had been burned? Would we never have known Galois theory? No. Someone else would have discovered it because it is inevitable.

Because it is simply true?

Yes. It’s a difficult philosophical question, to which we still don’t have the answer, but it’s an important question to be aware of. It’s not the same as the mathematics we use to calculate a tip—it goes to the heart of reality and of consciousness. It is all around us, with smart phones and computers and GPS devices and the algorithms that control our lives. The Amazon recommendations we are offered are based on very sophisticated algorithms, which analyse our past purchases, correlating us with other users. Mathematics is invading our world more and more and it communicates timeless, persistent and necessary truths which transcend time and space. The Langlands Programme should be as familiar to us as the theory of relativity.

What is the Langlands Programme?

Robert Langlands is a mathematician who occupies the office of Albert Einstein at the Institute of Advanced Studies in Princeton. His is one of the biggest ideas in the last 50 years. In the late 1960s he was able to find unexpected links between two fields of mathematics which seemed to be light years apart. One of them is number theory—this is something we can all relate to as it was driven by an impulse to understand the structure of numbers. We start with the simplest possible numbers 1,2,3,4, and so on, but then we realise that there are numbers that cannot be expressed as whole numbers or even as ratios of whole numbers, such as the square root of 2. It exists but doesn’t fit in the narrow framework. That’s one continent on this planet of mathematics. There are other continents—when we hear music we realise that the sound is composed of different instruments, of different notes. Each of those notes can be represented mathematically as a wave. In mathematics we call them harmonics and there is a deep theory that takes this further in which sophisticated signals can be deconstructed; this is called harmonic analysis.

These theories are as far apart as Europe and America but what the Langlands Programme does is enable transportation between any point in Europe to any point in America and back. Langlands discovered some secret patterns relating to numbers, which could be expressed using harmonics. That was the initial idea, but since then other mathematicians took this up and found similar and surprising patterns in geometry and quantum physics, in particular the electromagnetic duality. It may all sound very complicated, but in my book I explain [electromagnetic duality] using an analogy based on the recipe of borscht, which is a favourite soup in my home country, Russia.

Readers' comments

I count this as one of the most keepable Prospero pieces since Prospero's beginning, if not the . The Q's are unbelievably good, and the A's are unbelievably understandable. Mr. Frenkel must be a kindred soul to Feynman, who can explain anything in his field to any mind willing and curious to learn. Thanks and thanks and thanks again. This piece makes my day. No, it makes my entire holiday season. Will read the book and get more copies for my loved ones of all ages who is curious about the world we live in, and about how beauty exists and is not only for the poets.

I agree with the concept of teaching children to appreciate the depth and extent of mathematical concepts at as early an age as possible. However there is a great shortage of qualified mathematics teachers, in the United States and many areas of Europe, who would be capable of doing this. A large number of pupils are taught by teachers who are qualified in other subjects not necessarily related to mathematics. The result is that mathematics is taught by people with little real enthusiasm for the subject. A big factor in the teaching of mathematics is to ask appropriate "what if" questions designed to expand a child's horizons by posing questions which are not strictly in the syllabus. I was fortunate to have well qualified teachers who were able to do that. I shudder to think what would happen if the majority of third grade teachers were further burdened with teaching the basic concepts of quantum mechanics.

We would not expect good results if we were to trust our children's early education in English, to a teacher who was not a good English speaker. Yet all too often that standard of teaching is what happens in elementary mathematics.

My introduction to mathematics was somewhat different. As a small child I was given a thin book called "Quaint and Quick Methods in Arithmetic". The book contained a number of ways of simplifying mental arithmetic. I found the facility to easily add multiply divide and subtract numbers fascinating. Later on I progressed to geometry and algebra and went on to have a normal University mathematical education. The education and stood me in good stead and provided me with employment and entertainment for a lifetime.

Over time I formed the idea that there are three principal forms of communication, language, mathematics and art. Mathematics and art are two sides of the same coin. They are concerned largely with mentally developing models and their representation.

Language on the other hand has a somewhat less focused context. As far as education is concerned has been great concentration on literacy. Much more effort and funding is devoted to literacy, than is devoted to the corresponding quantitative methods of mathematics. As a result we have a great deal of emphasis in our daily lives on language. However the majority of people are not made sufficiently aware of languages deficiencies. Language is scarcely if ever exact. In addition there is a gray area in language, between what is said and what is meant. This area is a fruitful ground for politics, religion, law and philosophy. Politicians are well-known for obfuscation it is part of their stock in trade. Religious language is always open to interpretation and has given rise to a large number of sects within a given religion. The law, is also not as precise as it could be. And hence we have lawyers and judges whose job is to interpret it for us. I suppose this is not surprising since most laws are written by politicians. Philosophy is also not immune from the problems of language. Meanings can be lost in translation. It is also necessary to look at the added complication of idioms in language. I am also always amazed by the amount of effort that is put into reinterpreting the writings of philosophers from the classical period such as Plato. When one considers all the books and thousands of words that are being written on the subject of Plato you might be tempted to think that the contents of his relatively meager writings should by now have been exhaustively examined.

Mathematics has the advantage of being precise and not requiring extensive explanation providing you have a basic mathematical education. Art which consists of examining picture based models, perspectives, patterns, sounds and colors is closely allied to mathematics. It is however not as concerned with quantitative values as is mathematics.

One of the problems of educating the masses in mathematics, is that the job is often left to relatively uneducated instructors. Particularly in the early years when an interest in mathematics is first formed. The result is that many people remain quantitatively challenged throughout their lives. As a consultant, I spent many years in the 1960s and 70s trying to introduce statistics into medical research. I worked with a number of very highly educated doctors and was astounded by the general low level of their understanding of mathematics. Fortunately after many years of effort statistical examination of experimental and test results now seems to be well established in the medical profession. But it was not always so. On the other end of the spectrum I still regularly come across shop assistants who have difficulty in making change.

I think it is laudable to try and introduce the general populace to the beauty of mathematics, but it is also important to convince them of its utility. As a result of my early introduction to mental arithmetic even now when using computers and calculators I instinctively formulate what the end value of a series of calculations will be. It is not absolutely accurate but it is sufficiently accurate for me to go back and check if my formulation and the answers I get from the computer are different. I think if people were not as quantitatively challenged as they now are, the world might be a better and more honest place.

One cavil: I am incredibly tired of people asking about the beauty of mathematics. Really? You need to be convinced after fractal geometry? Or going back in time, after Kepler and then Newton figured the orbit of the planets and the rather amazing mathematically rendered truth that the earth falls toward the sun over and over.

The idea this needs to be brought up and explained is, I believe, part of the problem. As Mr. Frenkel says, you can explain difficult ideas in words and images ... but you can only do that if you take for granted that you can and the problem is we assume the contrary, that math is obscure and draws dreary images of conic sections. I believe Feynman said we should teach quantum to 10 year olds because their minds are flexible enough to understand. See? If we accept basics, like mathematics has phenomenal beauty, then we can actually start explaining and understanding difficult concepts rather than rely on the random book to entice a few readers beyond those who already get it.

I have enjoyed so much reading your post and the observations you made in it. Yes, the younger generation in America, in general, with large pockets of exceptions here and there, perhaps in great part due to the education available to K-12 citizens, are severely quantitatively challenged. I love and support your last sentence. It is bold but true, especially on the "honest" part.

I agree the system is the problem, but I thought it necessary to point out that because mathematicians can earn much more in industry and finance very few of them go into teaching. This means that teaching mathematics is more often than not, in the hands of people who are less qualified and enthusiastic than they could be. When my children were in school I had more than one occasion to correct elementary mistakes made by their teachers, there were also mistakes in some of their text books.

For my own part I had some success with children who were mathematically challenged by introducing them to probability. Most children quickly picked up how to assess the likelihood of something happening, based on the information they were given. I think the key is to introduce them to real life situations which are meaningful to them.

I have followed your conversation and find it fascinating. I invite myself to join. I hope that is all right.

@jomiku's observation regarding rote-learn would not be popularly received in the culture that is USA. But speaking as a person who came up from that method to learning, I can vouch for the merit of it. Let me, however, break down the "method" further.

I think one aspect fundament to any learning is information storage. For this aspect, nothing works better than rote-learn. Nor is there a shortcut to rote-learn. I note that these days, a lot of information is available at the click of a mouse. But what needs to be borne in mind is one needs to know enough about a subject field to be able to discern good (meaning accurate) from bad (meaning inaccurate). That is VERY IMPORTANT. The presumption that anyone can pick up anything about a subject and become an Einstein or a Justice Brandeis or a Vladimir Horowitz has led many a poor soul down a one-way road to nowhere and worse.

So rote-learning ACCURATE information is IMPORTANT, as later your entire learning life will depend on it.

BUT after that, rote-learning will not get one very far in the marathon that is the learning process.

CONCEPTS, which are contextual, and context, which is information-based, is the NEXT STEP.

Here is where the rote-learners fail to move on unless they UNDERSTAND the what and if and whereto and what for in the information they have rote-learned. Concepts are very difficult, whether it is about a point of technique in playing a musical instrument, or a point in set theory, or something one has memorized about what Wittgenstein said, or a poem Dylan Thomas wrote.

And here, at this stage of learning, is where something totally paradoxical happens. The quote is attributed to Albert Einstein: "Education is that which remains when one has forgotten everything learned in school".

Speaking from my personal experience, things really work the way Einstein said they work. You have to let go of what's been rote-learned in order to grasp the concept behind what you have rote-learned. This is very difficult to do. And this is also why rote-learn is sometimes a hindrance to onward learning (especially when what is rote-learned is bad information. Then that's a ship-wreck). This is why Richard Feynman refused to answer the Q posed by a lay journalist. The Q was: WHY do two magnets repel each other when their like poles are brought near each other? I wish I had the link to that youtube video (a poster led me to it about two months ago. I forgot which Blog). Feynman answered the Q, the answer of a true genius, by not answering it.

How does a teacher help a student make the leap from information to concepts? Well, those of us who have been fortunate to have great teachers in our lives know when a teacher has a passion for his/her subject (which almost always translates into he/she knows his/her stuff), this passion moves mountain. By "mountain" I mean the resistance to learning.

Resistance generally comes from fear. Fear generally comes from belief in the myths built and accrete around a subject field by "myth-builders" and "myth -spreaders" (both terms = euphemism for folks who start with inaccurate information as above explained, aka philistines ). Math is a good example of a myth-field in USA. Nobody fears math in China. Don't ask me who started the myths in US. I don't know.

Speaking of "myth-builders" and "myth-spreaders", most cultures have their own pet myth-field. For example, for the culture of religious fundamentalists, the myth-field is other faiths and other religions. The myth is you can't take a look at other faiths and other religions, for if you do, your own will crumble (which of course, is very dumb).

I digress. But the point is relevant to why I agree with @raggar's point as stated in his concluding sentence in the first post.

I write this in haste because of shortage of time. Please excuse typos and jagged writing where you find it.

It is, I think, really unfortunate that probability is introduced and then sort of disappears ... and very few people get to statistics (and fewer grasp it). Considering how the internet has made stuff like the shared birthday problem relatively well know ...

I have an old friend - through our kids - who works in math curriculum development. And I've been involved in numerous discussions about curriculum in our school district, which is one of the best.

And ...

I think we do it wrong. I think we spend way, way too much time trying to teach understandings of quantity and counting. We recognize these are difficult - empirically at least - but beat on them as though they're the key. I think rote learning of how to do things is better. Most countries outside the US either focus on rote learning or combine much more rote with degrees of understanding. Even testing in the US is set up to make students do poorly: we tend in math to test to identify what a student doesn't know rather than what that student can do. So we load them up with unfamiliar applications to see if they can "apply" concepts. Very few people can apply mathematical concepts.

I admit my perspective has been shaped by personal factors, along with exposure to curriculum debates. My own years spent working way ahead of grade, essentially on my own, and then being pulled into regular classes when I changed schools. That experience was shocking. And one of my children spent time in classes at a top high school in China so I developed an understanding of how that system works.

So in response, I'd say it's less the teacher than the way we teach. We ask people to inculcate understandings that are hard for the vast majority to grasp - ever. Because of that we miss the simpler path, which is to train the vast majority to do math with the understanding that relatively few will progress toward much higher levels of work and understanding. And you know what? That's kind of how we approach other subjects. In the old days, it was expected, for example, that every kid could play piano with the understanding that only a few could really play well.